# third “power” differential in vector calculus

What is the meaning of the following in vector calculus:

$$d^3\textbf{r}$$

where $\textbf{r}\in R^3$?

For example, it is used sometimes in the definition of the electric dipole moment (otherwise $d\tau$, the volume differential is used), as given in the first math line here:

http://en.wikipedia.org/wiki/Electric_dipole_moment#Expression_.28general_case.29

My thought was that it was the differential defined by the dot product of an infinitesimal vector with the cross product of two more infinitesimal vectors to make a parallelepiped, but I believe this vector volume differential already has a notation, but what is it?

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This notation is just shorthand for $dx\,dy\,dz$. It's not the greatest notation in the world as far as I'm concerned but it's certainly not the worst.